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In probability theory, the law of rare events or Poisson limit theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions. [1] The theorem was named after Siméon Denis Poisson (1781–1840). A generalization of this theorem is Le Cam's theorem
Thus, for example, if the uniform asymptotic negligibility (u.a.n.) condition is satisfied via an appropriate scaling of identically distributed random variables with finite variance, the weak convergence is to the normal distribution in the classical version of the central limit theorem.
Cameron–Martin theorem; Campbell's theorem (probability) Central limit theorem; Characterization of probability distributions; Chung–Erdős inequality; Condorcet's jury theorem; Continuous mapping theorem; Contraction principle (large deviations theory) Coupon collector's problem; Cox's theorem; Cramér–Wold theorem; Cramér's theorem ...
The free Poisson distribution [40] with jump size and rate arises in free probability theory as the limit of repeated free convolution (() +) as N → ∞. In other words, let X N {\displaystyle X_{N}} be random variables so that X N {\displaystyle X_{N}} has value α {\displaystyle \alpha } with probability λ N {\textstyle {\frac {\lambda }{N ...
A renewal process has asymptotic properties analogous to the strong law of large numbers and central limit theorem. The renewal function () (expected number of arrivals) and reward function () (expected reward value) are of key importance in renewal theory. The renewal function satisfies a recursive integral equation, the renewal equation.
A visual depiction of a Poisson point process starting. In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
Baron Siméon Denis Poisson (/ p w ɑː ˈ s ɒ̃ /, [1] US also / ˈ [invalid input: 'poison'] /; French: [si.me.ɔ̃ də.ni pwa.sɔ̃]; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electricity and magnetism, thermodynamics, elasticity, and ...
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate the corresponding electrostatic or gravitational ...